
To obtain an initial guess \(x^0\) , we considered the \(\ell _1\) -problem (REF ) as a quadratic program and applied MATLAB's quadprog as seen in [1]}, which required the split \(x = x_+ - x_-\)  with \(x_+,x_- \ge 0\) .
From the solution \((x_+^0, x_-^0)\)  we could recover \(x^0 = x_+^0 - x_-^0\) .
Note that in order to invoke the operator \(T_C\) , we now have to split \(x\)  into positive and negative part, because otherwise we do not have any nonnegativity constraints.
For \(T_C\)  we thus used \((x_+^0, x_-^0)\)  as is to initialize the algorithm.
Unfortunately, this split leads to a much higher computational cost for \(T_C\) , since in every Newton-step a system of \(6n + m\)  equations had to be solved.
